Bloch Theorem
The Bloch theorem, is a fundamental result in solid-state physics and the study of periodic systems. It describes the behavior of particles, such as electrons or other quantum particles, in a periodic potential, which is the case in crystalline solids.
The Bloch theorem states that in a periodic potential, the wavefunction of a particle can be written as the product of a plane wave and a function with the same periodicity as the potential. Mathematically, the theorem can be expressed as:
\psi(\mathbf r) = e^{ik \mathbf r} u(\mathbf r)
where:
- \psi(\mathbf r) is the wavefunction of the particle
- k is the wave vector (also known as the crystal momentum or Bloch wave vector)
- \mathbf r is the position vector
- e^{ik \mathbf r} is a plane wave
- u(\mathbf r) is a periodic function with the same periodicity as the potential, known as the Bloch function or periodic part
The Bloch function u(\mathbf r) has the property that u(\mathbf r) + \mathbf R, where \mathbf R is any Bravais lattice vector of the crystal. This means that the Bloch function has the same periodicity as the crystal lattice.
The Bloch theorem has several important consequences and implications:
- Energy band structure: The Bloch theorem leads to the concept of energy bands in solids. The allowed energy levels for electrons in a crystal are organized into bands, separated by energy gaps (forbidden energy regions). This band structure determines the electronic properties of the material, such as conductivity, semiconductivity, or insulating behavior.
- Effective mass: The Bloch theorem allows for the definition of an effective mass for electrons or holes in a periodic potential, which is different from the free-particle mass. The effective mass determines the behavior of charge carriers in the material and is crucial for understanding transport properties.
- Boundary conditions: The Bloch theorem provides the appropriate boundary conditions for solving the Schrödinger equation in a periodic potential, which is essential for calculating the electronic structure of crystalline solids.
- Brillouin zones: The wave vector k in the Bloch theorem is defined within the first Brillouin zone, which is a fundamental region in reciprocal space. The properties of the Bloch waves are periodic in k-space, allowing for the study of the electronic structure within a single Brillouin zone.
The Bloch theorem is a cornerstone of solid-state physics and is widely used in the study of electronic, optical, and transport properties of crystalline materials. It provides a powerful theoretical framework for understanding the behavior of particles in periodic systems and has numerous applications in condensed matter physics, materials science, and related fields.
Periodic boundary conditions
Lets’ consider a system defined on a segment [0,L) in one dimension. Under periodic boundary conditions:
- A particle that moves past x=L reenters the system at x=0.
- Similarly, a particle crossing x=0 in the opposite direction appears at x=L.
Assuming that the points are equally spaced by a distance a, so that there are N = \frac{L}{a} points this can be expressed as:
x = x + m a
where m is any integer, indicating that the system’s configuration repeats every N units. This relation effectively creates a topological circle, despite the linear nature of the physical space. In one dimension can be represented as:
So, moving along m times, the potential will be the same:
V(x + ma) = V(x)
This will be true even if m \gg N, as it will be just equivalent to go around the topological circle many times. If the chain is very long, then this circle will not be too different from an infinite long chain, as the bend required to be periodic will not be noticeable.
The problem has been changed from dealing with an infinite system with a finite system which shows a periodic behavior.
Proof
Consider a one-dimensional periodic potential V(x) with a period a, such that V(x + ma) = V(x) for all x. The time-independent Schrödinger equation for an electron in this potential is given by:
\left[-\frac{\hbar^2}{2m} \frac{\mathrm d^2}{\mathrm dx^2} + V(x)\right] \psi(x) = E \psi(x)
Now, let’s define a translation operator T(a) such that T(a)\psi(x) = \psi(x + a). Since the potential is periodic, we can write:
V(x + a) = V(x)
Applying the translation operator T(a) to both sides of the Schrödinger equation, we get:
\left[-\frac{\hbar^2}{2m} \frac{\mathrm d^2}{\mathrm dx^2} + V(x)\right] T(a)\psi(x) = E T(a)\psi(x)
Substituting T(a)\psi(x) = \psi(x + a), we obtain:
\left[-\frac{\hbar^2}{2m} \frac{\mathrm d^2}{\mathrm dx^2} + V(x)\right] \psi(x + a) = E \psi(x + a)
Comparing this equation with the original Schrödinger equation, we see that \psi(x + a) must also be a solution with the same energy E.
Now, let’s assume that \psi(x) is a solution to the Schrödinger equation with energy E. We can express \psi(x + a) as:
\psi(x + a) = \psi(x) e^{ika}
where k is a constant (to be determined).
Substituting this into the Schrödinger equation for \psi(x + a), we get:
\left[-\frac{\hbar^2}{2m} \frac{\mathrm d^2}{\mathrm dx^2} + V(x)\right] \psi(x) e^{ika} = E \psi(x) e^{ika}
Dividing both sides by e^{ika}, we obtain:
\left[-\frac{\hbar^2}{2m} \frac{\mathrm d^2}{\mathrm dx^2} + V(x)\right] \psi(x) = E \psi(x)
This is the original Schrödinger equation for \psi(x). Therefore, the wavefunction \psi(x) can be written as:
\psi(x) = e^{ikx} u(x)
where u(x) = u(x + a) is a periodic function with the same periodicity as the potential V(x).
This is the Bloch theorem, which states that in a periodic potential, the wavefunction can be expressed as the product of a plane wave (e^{ikx}) and a periodic function (u(x)) with the same periodicity as the potential.
The proof can be extended to three dimensions and more general crystal structures by using the translation symmetry of the periodic potential and the translation operators along different lattice vectors.